The full reasoning can be found below the Sudoku.
May 18 - Super Hard
Puzzle Copyright © Kevin Stone
Reasoning
R3C8 is the only square in row 3 that can be <6>
R6C7 is the only square in row 6 that can be <2>
R9C5 is the only square in row 9 that can be <1>
R1C7 is the only square in row 1 that can be <1>
R5C8 is the only square in row 5 that can be <1>
R5C9 is the only square in row 5 that can be <5>
R3C6 is the only square in row 3 that can be <5>
R3C2 is the only square in row 3 that can be <2>
R4C3 is the only square in row 4 that can be <5>
R4C2 is the only square in row 4 that can be <9>
R2C3 is the only square in row 2 that can be <9>
R7C2 is the only square in row 7 that can be <1>
R7C8 is the only square in row 7 that can be <5>
R2C7 is the only square in row 2 that can be <5>
R7C4 is the only square in row 7 that can be <6>
R8C8 is the only square in row 8 that can be <9>
R9C9 is the only square in row 9 that can be <2>
R9C1 is the only square in row 9 that can be <9>
R5C5 is the only square in column 5 that can be <6>
R3C4 is the only square in block 2 that can be <3>
Squares R2C2 and R5C2 in column 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <48>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the column.
R6C2 - removing <48> from <3468> leaving <36>
R8C2 - removing <48> from <3468> leaving <36>
Squares R5C1 and R5C2 in block 4 form a simple naked pair. These 2 squares both contain the 2 possibilities <48>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the block.
R6C3 - removing <48> from <3468> leaving <36>
Intersection of column 9 with block 3. The value <7> only appears in one or more of squares R1C9, R2C9 and R3C9 of column 9. These squares are the ones that intersect with block 3. Thus, the other (non-intersecting) squares of block 3 cannot contain this value.
R2C8 - removing <7> from <478> leaving <48>
Squares R2C2 and R2C8 in row 2 form a simple naked pair. These 2 squares both contain the 2 possibilities <48>. Since each of the squares must contain one of the possibilities, they can be eliminated from the other squares in the row.
R2C4 - removing <8> from <178> leaving <17>
R2C5 - removing <8> from <278> leaving <27>
R2C6 - removing <8> from <1278> leaving <127>
R1C5 is the only square in block 2 that can be <8>
Intersection of column 3 with block 7. The value <8> only appears in one or more of squares R7C3, R8C3 and R9C3 of column 3. These squares are the ones that intersect with block 7. Thus, the other (non-intersecting) squares of block 7 cannot contain this value.
R7C1 - removing <8> from <348> leaving <34>
Squares R6C2, R6C3, R8C2 and R8C3 form a Type-1 Unique Rectangle on <36>.
R8C3 - removing <36> from <3468> leaving <48>
Squares R2C5, R8C5, R2C6 and R8C6 form a Type-4 Unique Rectangle on <27>.
R2C6 - removing <7> from <127> leaving <12>
R8C6 - removing <7> from <2378> leaving <238>
Intersection of column 6 with block 5. The value <7> only appears in one or more of squares R4C6, R5C6 and R6C6 of column 6. These squares are the ones that intersect with block 5. Thus, the other (non-intersecting) squares of block 5 cannot contain this value.
R4C4 - removing <7> from <1478> leaving <148>
R6C4 - removing <7> from <478> leaving <48>
Squares R7C1 (XY), R7C6 (XZ) and R8C3 (YZ) form an XY-Wing pattern on <8>. All squares that are buddies of both the XZ and YZ squares cannot be <8>.
R8C4 - removing <8> from <78> leaving <7>
R8C6 - removing <8> from <238> leaving <23>
R8C5 can only be <2>
R2C4 can only be <1>
R8C6 can only be <3>
R2C5 can only be <7>
R8C2 can only be <6>
R7C6 can only be <8>
R2C6 can only be <2>
R7C9 can only be <4>
R6C6 can only be <7>
R7C1 can only be <3>
R1C9 can only be <7>
R8C7 can only be <8>
R6C2 can only be <3>
R9C3 can only be <8>
R8C3 can only be <4>
R4C7 can only be <4>
R9C7 can only be <6>
R3C9 can only be <8>
R3C1 can only be <7>
R2C8 can only be <4>
R4C4 can only be <8>
R6C3 can only be <6>
R6C8 can only be <8>
R4C6 can only be <1>
R6C4 can only be <4>
R4C8 can only be <7>
R1C1 can only be <4>
R1C3 can only be <3>
R5C1 can only be <8>
R2C2 can only be <8>
R5C2 can only be <4>
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